answer: To find all answers to the equation 2cos(x) + 1 = sec(x), we can use the following steps:
Rewrite sec(x) as 1/cos(x), using the definition of secant.
Multiply both sides by cos(x), to eliminate the fraction.
Simplify and rearrange the terms to get a quadratic equation in cos(x).
Solve the quadratic equation using the quadratic formula or factoring, if possible.
Find the values of x that satisfy the equation, using the inverse cosine function and the periodicity of cosine.
Here are the steps in detail:
2cos(x) + 1 = sec(x)
2cos(x) + 1 = 1/cos(x)
2cos^2(x) + cos(x) - 1 = 0
(2cos(x) - 1)(cos(x) + 1) = 0, by factoring
cos(x) = 1/2 or cos(x) = -1, by setting each factor to zero
x = cos^-1(1/2) or x = cos^-1(-1), by taking the inverse cosine of both sides
x = π/3 + 2πn or x = -π/3 + 2πn or x = π + 2πn, where n is any integer, by using the inverse cosine function and the periodicity of cosine
Therefore, the general solutions are:
x = π/3 + 2πn x = -π/3 + 2πn x = π + 2πn